**Effects of Boundary Conditions on Postbuckling Strengths **

**of Composite Laminate with Various Shaped Cutouts **

**under In-Plane Shear **

**Dinesh Kumar1 _{, Shamsher Bahadur Singh}2***

1_{Department of Mechanical Engineering, Malaviya National Institute of Technology (MNIT), Jaipur, India; }2_{Department of Civil }
Engineering, Birla Institute of Technology and Science (BITS), Pilani, India.

Email: *_{sbsingh@pilani.bits-pilani.ac.in }

Received January 16th_{, 2013; revised February 10}th_{, 2013; accepted March 10}th_{, 2013 }

Copyright © 2013 Dinesh Kumar, Shamsher Bahadur Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**ABSTRACT **

Cutouts are often provided in composite structural components for practical reasons. For instance, aircraft components
such as wingspar, fuselage and ribs are provided with cutouts for access, inspection, fuel lines and electric lines or to
reduce the overall weight. This paper addresses the effect of boundary condition on buckling and postbuckling re-
sponses, failure loads, and failure characteristics of composite laminate with various shaped cutouts (*i.e.*, circular,

square, diamond, elliptical-vertical and elliptical-horizontal) and having different lay-ups under in-plane shear (positive and negative) load, using finite-element method. The FEM formulation is based on the first order shear deformation theory in conjunction with geometric nonlinearity using von Karman’s assumptions. The 3-D Tsai-Hill criterion is used to predict the failure of a lamina while the onset of delamination is predicted by the interlaminar failure criterion. It is observed that the effect of boundary condition on buckling, first-ply failure and ultimate failure loads of a quasi-iso- tropic laminate with cutout is more for positive shear load than that for the negative shear load for almost all cutout shapes. It is also noted that under in-plane shear loads postbuckling stiffness of (0/90)4s laminate with circular cutout is maximum, while it is minimum for (45/−45)4s laminate with circular cutout, irrespective of boundary conditions.

**Keywords:** Boundary Conditions; Buckling; Postbuckling; Cutouts; Composite Laminate; FEM

**1. Introduction **

Currently, thin composite laminated panels are used in
almost all modern advanced engineering applications such
as spacecraft, high speed aircrafts, naval vessels and au-
tomobiles because of their superior specific properties
(*i.e.*, stiffness-to-weight and strength-to-weight ratios) as

compared to their metallic counterparts. Further, cutouts are often provided in these panels for practical considera- tions. For instance, cutouts in wing spars and cover pan- els of commercial transport wings and military fighter wings are provided to form ports for mechanical and ele- ctrical systems, damage inspection, fuel lines, and also to reduce the overall weight of the composite structure. The presence of these cutouts forms free edges in the com- posite laminates under various loading and boundary con- ditions, which in turn cause high interlaminar stresses [1] leading to loss of stiffness and premature failure of la-

minate owing to delamination. These composite structu- ral laminates with cutouts are required primarily to resist buckling, and they must carry a load well into the post- buckling range to yield weight savings by utilizing its reserve strength beyond buckling. Further, boundary con- ditions affect the buckling and postbuckling behavior of laminated panels under various loading conditions such as in-plane compression, shear, and combined inplane shear and compression [2-6]. In addition, the shear directions also affect the postbuckling behavior of composite plates [7,8]. Thus, it is imperative to have thorough knowledge of buckling, postbuckling, failure characteristics and streng- th of thin composite panels with cutouts under various edge boundary conditions subjected to in-plane shear (po- sitive and negative) loads, for their cost effective and ef- ficient design.

A comprehensive review on buckling and postbuck- ling behavior of laminated composite plates with a cutout by Nemeth [9] has shown that only limited works [10-13]

are available on buckling and postbuckling behavior of
laminated composite panels with cutouts under shear
loads for different boundary conditions. Moreover, these
studies are primarily for laminates with central circular
cutouts. Thereafter, Jha and Kumar [14] studied the first-
ply failure of symmetric square laminates (without cut-
outs) under the combined effect of in-plane and trans-
verse loads using FEM. Jain and Kumar [15] examined
the effect of cutout shape, size and the alignment of the
elliptical cutout on the buckling and the first-ply failure
loads of laminates under uni-axial compression. Guo [16]
conducted numerical and experimental studies to inves-
tigate the effect of reinforcements around cutouts on the
stress concentration and buckling behaviour of a carbon/
epoxy composite panel with simply-supported and clamp-
ed boundary conditions under in-plane shear load. Guo *et *
*al.* [17] examined the effect of cutout and various edge

reinforcements in a composite C-section beam under sta- tic shear load and demonstrated that the cutout induced stress concentration can be reduced significantly by ap- propriate cutout shape and edge reinforcements. Very late- ly, Kumar and Singh [18] studied the buckling and post- buckling response and strengths of simply-supported com- posite laminate with various shaped cutouts under in- plane shear.

From the above literature review, it is manifested that
there are no investigations on the effects of boundary con-
ditions on buckling and postbuckling behavior of com-
posite laminates with non-circular cutouts under in-plane
shear loading conditions. Furthermore, most of the avai-
lable studies on shear buckling and postbuckling of la-
minates with cutouts under given boundary conditions
are mainly concerned with the study of load versus out-of-
plane displacement relationship at loads beyond buckling
without commenting on their actual reserve strength in
the postbuckling range. The aim of the present investiga-
tion is to explore the effect of boundary condition on
buckling and postbuckling responses, failure loads and
failure characteristics of a quasi-isotropic [*i.e.*, (+45/−

45/0/90)2s] laminate with central circular as well as non-
circular (*i.e.*, square, diamond, elliptical) cutouts under

in-plane positive and negative shear loads. In addition,
for various boundary conditions, the effect of the com-
posite lay-up on buckling and postbuckling characteris-
tics of the laminate with a circular cutout is also studied
by taking three most practical laminate configurations,
namely, quasi-isotropic [*i.e.*, (+45/−45/0/90)2s], angle-ply
[*i.e.*, (45/−45)4s], and cross-ply [*i.e.*, (0/90)4s].

**2 Present Study **

**2.1. Finite Element Formulation **

A special-purpose computer program is developed to

carry out the study which is based on the finite-element formulation using the first-order shear deformation the- ory with a nine-noded Lagrangian element having five degrees of freedom per node. Geometric non-linearity based on von Karman’s assumptions has been incorpo- rated. The formulation is based on the virtual work equa- tion for a continuum in the total Lagrangian coordinate system under the assumption of small strains.

The displacement within an element is interpolated by an expression of the form

###

T###

###

###

5 1

, , , , *n*

*x* *y* *i* *i*

*U* _{}*u v w* _{}

###

_{}

*N I*

*a*;

_{i}where

###

*U*is the value of displacement components at

a point within an element; *n* the number of nodes in an

element; *Ni* the shape functions of a nine noded Lagran-

gian element, for *i* = 1, *n*; *I*5 the 5 × 5 unit matrix;

###

T0, 0, 0, ,

*i* *i* *i* *i* *xi* *yi*

*a* _{}*u* *v* *w* _{} is the nodal displacement
vector for *i*th node.

From the principle of virtual work and the total La- grangian formulation, the element nonlinear equilibrium equation is derived as:

###

###

T T T0 *b* *s* d 0

*A*

*a*

*B* *N* *B* *M* *B* *Q* *A* *F*

_{}

_{}

_{}

where

###

*a*is the residual force which is a function of

displacement vector

###

*a*;

###

*B*0 ,

*Bb*and

###

*B*

###

*s* the strain-

displacement matrices corresponding to in-plane axial,
bending and shear strains, respectively; *N*

###

the stress re-sultants per unit length;

###

*M*the moment resultants per

unit length;

###

*Q*the transverse shear stress resultants per

unit length; *F *the external applied loads (includes in-plane

loads as well as transverse forces).

The Newton-Raphson method is used to solve these nonlinear algebraic equations using a combined incre- mental and iterative procedure. If for an initial estimate of

###

*aj*(

*i.e.*, for

*j*

th_{ iteration), the residual forces }

###

*a*0,

* _{j}* then an improved solution

###

*aj*

_{1}is ob-

tained by equating to zero the linearized Taylor’s series expansion of

###

_{1}

*j*

*a*

_{} in the neighborhood of

###

*a*as

_{j}###

_{1}

###

_{T}###

*0*

_{j}*j* *j*

*a* *a* *K* *a*

_{}

where

###

*aj*is the incremental displacement vector

and *KT* is the tangent stiffness matrix evaluated at

###

*aj*

and is given by:

###

###

*j*

*T*

*a*
*K*

*a*

The improved solution is then found as:

###

*aj*1

*aj*

*aj*.

gence of the solution, the load is applied in small incre- ments. The iterative solution is checked for convergence using the following criterion:

1 2 T

T 100

*F F*

_{}

where is sufficiently small number, *i.e.*, 0.001%.

The integration of expressions for and *KT* is

carried out using the Gaussian quadrature. A selective in- tegration scheme is adopted with integration rule to evaluate integrals of the functions of the membrane and the bending behavior and integration rule is used for the transverse shear component.

###

*a*

3 3

2 2

**2.2. Failure Model and Definition of Failure **

Failure of a lamina is predicted by tensor polynomial form of the 3-D Tsai-Hill criterion [19], [see Appendix (1)], wherein five stress components in material direc- tions (three in-plane stresses and two transverse shear stresses) were calculated at mid thickness of each layer of individual element using the constitutive equations and by applying proper transformation. In addition, an attempt has been made in the present study to predict the onset of delamination at interface of two adjacent layers using interlaminar failure criterion [20], [see Appendix (2)]. Three transverse stresses at each gauss point on the corresponding interface are calculated in material direc- tions using integration of equilibrium equations and by applying proper transformation. The in-plane stress va- riations used in each equilibrium equation are derived from nodal values of in-plane stresses. To predict the ul- timate failure of laminate, a progressive failure procedure as used by Singh and Kumar [8] has been implemented. In this progressive failure procedure, at each load step, gauss point stresses are used in tensor polynomial form of the Tsai-Hill failure criterion. If failure occurs at a gauss point in a layer of an element, a reduction in the appropriate lamina stiffness is introduced in accordance with the mode of failure. The laminate stiffness is recom- puted and failure is checked again at the same load step. If no failure occurs, the process is repeated at next load step. Ultimate failure is said to have occurred when the onset of delamination occurs at interface of any two lay- ers of any element or when the plate is no longer able to carry any further increase in load due to large transverse deflection.

**2.3. Material Properties and Geometric Model **

Properties of the material (T300/5208 graphite epoxy) of
each lamina are presented in **Table 1**. A full square
laminate of size 279 mm 279 mm × 2.16 mm with ply
thickness 0.135 mm is considered. A quasi-isotropic

laminate, having stacking sequence (+45/−45/0/90)2s (*i.e.*,
total 16 layers, bottom layer being the first layer), with
and without a central cutout of various shapes (*i.e.*,

square, circular, diamond, elliptical-vertical and ellipti-
cal-horizontal) has been investigated. As reported by
Kumar and Singh [18], that the effect of cutout shapes on
buckling and postbuckling responses of the quasi-iso-
tropic laminate is significant for large cutout sizes. Keep-
ing this in view, a laminate having cutout of area A3 (as
designated by Kumar and Singh [18]) are considered in
the present study. The area A3 is equal to the area of the
square cutout having aspect ratio (*i.e.*,* c/b*, where *c* is the

size of square cutout and *b* is width of square laminate)

equal to 0.42. Details of the cutout shapes and their di-
mensions are given in **Table 2**. In addition, for various
boun-dary conditions, the effects of composite lay-ups
[*i.e.*, (+45/−45/0/90)2s, (45/−45)4s and (0/90)4s] on buck-
ling and postbuckling responses of the laminate with a
typical circular cutout of size A3 are also investigated.

**2.4. Boundary and Loading Conditions **

Three different types of flexural edge boundary condi-
tions, namely BC1, BC2 and BC3 (as shown in **Figure 1**)
are considered; BC1 refers to a plate with all edges sim-
ply supported, BC2 refers to a plate with two longitudi-
nal edges (*y = *0 and *y* = *b*) simply supported and the

**Table 1. Material Properties of T300/5208 (pre-peg) graph- **
**ite-epoxy. **

Mechanical properties Values Strength properties Values

*E*1 132.58 GPa *Xt * 1.52 GPa

*E*2 = *E*3 10.80 GPa *Xc* 1.70 GPa

*G*12* = G*13 5.70 GPa *Yt = Zt* 43.80 MPa

*G*23 3.4 GPa *Yc = Zc* 43.80 MPa

12 = 13 0.24 *R * 67.60 MPa
23 0.49 *S = T * 86.90 MPa

**Table 2. Details of cutout shapes and their dimensions. **

Cutout shape Ratio* _{Cutout size for cutout area A}
3

Square c/b 0.420

Circular d/b 0.474

Diamond c/b 0.420

e/b 0.335 Elliptical-vertical

f/b 0.670

e/b 0.670 Elliptical-horizontal

f/b 0.335

BC1 (a)

BC2 (b)

BC3 (c)

**Figure 1. Details of various boundary conditions: (a) BC1, **
**(b) BC2, and (c) BC3; with the shear load directions. **

other two edges clamped (*i.e.*, *x= *0 and *x = b*) and BC3

refers to a plate with all edges clamped. In all three cases,
the in-plane boundary conditions on edges *x= *0, *x = b*, *y *
*= *0 and *y* = *b* related to in-plane displacements in *x*- and
*y*-direction (*i.e.*, *u* and *v*, respectively) are identical.

In-plane uniformly distributed shear loads (positive and
negative) are applied on all four edges of the laminate by
taking equivalent nodal forces at boundary edge of each
boundary element. The notations for positive and nega-
tive shear loads are also depicted in **Figure 1**.

Results for failure loads and the corresponding deflec- tions are presented in the following non-dimensionalized forms:

In-plane shear load: 2 3 2

*xy*

*N b* *E h* * *

Maximum transverse deflection: *w*max*/h*

Here, *E*2 is the transverse elastic modulus of a lamina;

*h* is the thickness of the laminate; *b* is the width of the

square plate; *Nxy* is thein-plane shear loads per unit width

of the plate; and, *w*max is the maximum transverse deflect
tion.

**2.5. Convergence Study **

To fix the number of elements in the finite element mesh,
a convergence study was conducted for a simply-sup-
ported quasi-isotropic laminate with a central square
cutout of area A3 using 72, 96, 120, 144 and 168 ele-
ments. The convergence of buckling and first-ply failure
loads was checked under positive shear load. Results of
convergence study are shown in **Table 3**. From **Table 3**

it can be observed that buckling and first-ply failure loads do not show any variation after 144 numbers of elements and hence convergence is achieved for 144 elements. For the sake of uniformity, finite element mesh of 144 ele- ments has been considered for the laminate with all shaped cutouts. In the case of laminate without cutout, the convergence of results has been obtained for finite ele- ment mesh of 5 × 5 and corroborate with the results ob- tained by Singh [20]. Schematic of finite element meshes along with element- and node-numbering schemes for a typical square laminate without cutout and with a circular

**Table 3. Convergence study. **

Nos. of elements

Nos. of nodes

Non-dimensionalized
buckling load
(*i.e*., *Nxyb*2*/E*2*h*3)

Non-dimensionalized
first-ply
failure load
(*i.e.*, *Nxyb*

2

*/E*2*h*3)

72 336 19.5248 32.7201

96 432 19.4175 37.5476

120 528 19.4175 37.6549

144 624 19.4175 37.7622

cutout are illustrated in **Figures 2** and **3**, respectively.
Schematics of finite element mesh for square laminates
with other cutout shapes are shown in **Figure 4**.

(*i.e.*, circular, square, diamond, elliptical-vertical and

elliptical-horizontal, respectively) for both directions of
shear load. Points corresponding to first-ply failure and
ultimate failure are also highlighted in **Figures 6** and

**7(a)-(e)**. Details of failure characteristics for the laminate
without and with various shaped cutouts subjected to dif-
ferent boundary conditions are tabulated in **Tables 5** and
6 for positive and negative shear loads, respectively. The
buckling load of the laminate with a cutout of various
shapes is less than that of the laminate without cutout, ir-
respective of edge boundary conditions and directions of

**3. Verification of Results **

The accuracy of the developed program under in-plane
shear is checked by comparing results (buckling loads,
postbuckling response and first-ply failure loads) with
the numerical and/or experimental results published by
Jha and Kumar [14], Guo [16] and Kosteletos [21].** Ta- **

**ble 4** contains the details of comparison along with the

validated results. The material properties used were the
same as given in the respective references. From** Table 4**,
a good agreement of the results from the developed pro-
gram can be observed with the results presented by Jha
and Kumar [14], Guo [16] and Kosteletos [21]. Further,
the load-deflection response of (45/−45)2s laminate with-
out cutout obtained in the present study matches with that
of Kosteletos [21], as shown in **Figure 5**.

1 2 3 4 5 6 7 8 9 10 1

12 13 14 15 16 17 18 19 20 21 22

23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52 53 54 55

67 68 69 70 71 72 73 74 75 76 77

56 57 58 59 60 61 62 63 64 65 66

78 79 80 81 82 83 84 85 86 87 88

89 90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

1

1 6

5 4

3 2

10 9

8 7

Node Number

Element Number

13 12

11 14 15

17

16 18 19

21

20 25 24

22 23

1 2 3 4 5 6 7 8 9 10 1

12 13 14 15 16 17 18 19 20 21 22

23 24 25 26 27 28 29 30 31 32 33

34 35 36 37 38 39 40 41 42 43 44

45 46 47 48 49 50 51 52 53 54 55

67 68 69 70 71 72 73 74 75 76 77

56 57 58 59 60 61 62 63 64 65 66

78 79 80 81 82 83 84 85 86 87 88

89 90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109 110

111 112 113 114 115 116 117 118 119 120 121

1

1 6

5 4

3 2

10 9

8 7

Node Number

Element Number

13 12

11 14 15

17

16 18 19

21

20 25 24

22 23

**4. Results and Discussions **

**4.1. Effects of Cutout Shapes for Various **
**Boundary Conditions **

The effects of edge boundary conditions (BC1, BC2 and
BC3) on load-deflection responses of the symmetric
quasi-isotropic laminate, (+45/−45/0/90)2swithout cutout
are shown in **Figure 6**, for positive and negative shear
loads. **Figures 7(a)-(e)** depict the load-deflection re-

sponses of the laminate with a cutout of various shapes **Figure 2. Meshing of a square laminate without cutout showing element- and node-numbering scheme. **

** **

(a) (b)

(c) (d)

**Figure 4. Meshing of square laminate with: (a) Diamond; (b) **
**Square; (c) Elliptical-horizontal; and (d) Elliptical-vertical **
**cutouts. **

0 1 2 3

0 20 40 60 80 100 120 140

4

**Nxy**

**b**

**2** **/E**

**2**

**h**

**3**

**w _{max}/h**

Present study (+ve shear) Kosteletos [21] (+ve shear) Present study (-ve shear) KKosteletos [21] (-ve shear)

**Figure 5. Comparison of load-deflection response of (45/ **

**−45)2s laminate without cutout with Kosteletos [21] under **

**positive and negative shear loads. **

shear load. For positive shear load, the maximum and mi- nimum reductions of buckling loads of the laminate with a cutout (except for the laminate with a diamond cutout under BC3 boundary condition) as compared to the la- minate without cutout are observed to be for BC1 and BC3 boundary conditions, respectively, for all cutout shapes. Variations in buckling load with cutout shape are within 11.9%, 12.8% and 4.74% for boundary conditions BC1, BC2 and BC3, respectively, in the case of positive

0 1 2 3 4 5 6 7 8 9

0 20 40 60 80 100 120 140 160 180 200 220

First-ply failure Ultimate failure

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**

**b**

**2/E**

**2**

**h**

**3**

**w _{max}/h**

**Figure 6. Effects of boundary conditions (BC1, BC2 and **
**BC3) on load-deflection responses of quasi-isotropic lami- **
**nate without cutout under positive and negative shear loads. **

shear load. However, for negative shear load, there is no such definite prediction of change in buckling load of the laminate with a cutout as compared to the laminate with- out cutout for different boundary conditions. Also, under negative shear load, the variations in buckling load with cutout shape are within 15.8%, 16.4% and 21.4% for boun- dary conditions BC1, BC2 and BC3, respectively. So, it can be observed that the effect of cutout shapes on buck- ling load is more for BC2 and BC3 boundary conditions under positive and negative directions of shear load, re- spectively.

Further, for both directions of shear load and all cutout
shapes other than diamond, the laminate with clamped
boundary condition (BC3) has higher buckling load as
compared to the laminate with other two types of
bound-ary conditions (*i.e.*, BC1 and BC2), as given in **Tables 5 **

and **6**. It is also to be noted from **Figure 7(c)** that the
laminate with a diamond shape cutout under BC3 boun-
dary condition shows no buckling failure for both direc-
tions of shear load because of early delamination of the
laminate before buckling.

**Table 4. Verification of results. **

S. No. Reference Laminate/stacking _{sequence } _{conditions }Boundary _{conditions }Loading Result validated In present _{study } In reference

Non-dimensionalized first-ply

failure load *i.e.*, *Nxyb*2/*E*2*h*3 61.3 58.9

+ve shear

*w*max/*h* at first-ply failure load 1.40 1.27

Non-dimensionalized first-ply

failure load *i.e.*, *Nxyb*2/*E*2*h*3 81.9 82.6

1 _{Kumar}Jha and a_{ (2002) }

479.88 mm × 479.88 mm × 3.72 mm plate without

cutout/(±45/0/90)2s

Simply supported on all

edges

−ve shear

*w*max/*h* at first-ply failure load 2.39 2.42

Simply supported on all edges

+ve shear Buckling load (kN) 8.54 8.40 (8.50)b

2 Guo (2007)

320.0 mm × 320.0 mm × 2.0 mm plate with central circularcutout of

diameter44 mm

(*i.e.*, d/b = 0.137)/(±45)4s _{on all edges}Clamped +veshear Buckling load (kN) 11.6 11.4

Non-dimensionalized

buckling load *i.e.*, *Nxyb*2/*E*2*h*3 63.0 63.3

+ve shear

*w*max/*h* at *Nxyb*2/*E*2*h*3 = 70.0 1.66 1.98

Non-dimensionalized

buckling load* i.e.*, *Nxb*2/*E*2*h*3 107.7 106.7

3 Kosteletos_{(1992) }c

279.0 mm × 279.0 mm × 2.16 mm plate without

cutout/(±45)2s

Clamped on all edges

−ve shear

*w*max/*h* at *Nxyb*2/*E*2*h*3 = 110.0 1.10 1.25
a_{First-ply failure results are based on Tsai-Hill failure criterion. }b_{The quantities inside and outside parentheses represent the critical buckling load (kN) from the }

FE analysis and experimental investigation, respectively. c_{Kosteletos’s results presented here are extracted from the figure given in the paper. }

**Table 5. Details of failure characteristics of quasi-isotropic laminate with and without various shaped cutouts for boundary **
**conditions BC1, BC2 and BC3 under positive shear load. **

Boundary condition

BC1 BC2 BC3

Cutout shape

BLa_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/ }

mode of FPF

UFf

load/mode of UF

BLa_{/FPF}b

load/
(*w*max/*h*)c

FEd_{/FL}e_{/ }

mode ofFPF

UFf_{ load/mode }

of UF

BLa_{/FPF}b_{ }

load/
(*w*max/*h*)c

FEd_{/FL}e_{/mode of}

FPF

UFf

load/mode of UF

Without

cutout 43.7/61.4/1.41 1/1/

transverseg _{of stiffness }151.4/loss 65.4/87.0/1.40 _{transverse}1/1/ g 150.4/loss of _{stiffness } 83.7/102.7/1.24 1/1/transverseg 159.7/loss of _{stiffness }

Circular 20.7/39.3/2.07 _{transverse }139/1/ _{of stiffness }101.9/loss 32.7/56.5/2.03 _{transverse }1/16/ 132.1/loss of _{stiffness } 43.9/69.2/1.85 133/1/transverse 119.0/loss of _{stiffness }

Square 19.3/37.8/2.02 _{transverse }139/16/ _{delamination }64.5/ 31.2/49.5/1.80 _{transverse }144/1/ _{delamination}73.2/ 42.2/52.1/1.15 144/1/transverse _{delamination}78.9/

Diamond 21.9/39.7/2.14 _{transverse }139/1/ _{delamination }40.3/ 34.4/36.4/0.56

90/interface of layers 15 & 16/delamination

36.4/

delamination -/37.3/-

90/interface of layers 15 & 16/delamination

37.3/ delamination

Elliptical

vertical 19.3/37.9/2.12 139/1/ transverse

44.5/

delamination 30.0/50.1/1.84

48/ interface of layers 15 & 16/delamination

50.1/

delamination 43.0/54.4/1.25

48/interface of layers 15 & 16/delamination

54.4/ delamination

Elliptical

horizontal 19.7/38.2/2.11 transverse 139/1/ delamination 43.1/ 32.9/52.8/1.91

96/interface of layers 15 & 16/delamination

52.8/

delamination 44.3/57.1/1.34

96/interface of layers 15 & 16/delamination

57.1/ delamination

a_{Buckling load; }b_{First-ply failure; }c_{Non-dimensionalized maximum transverse deflection at the first-ply failure; }d_{First failed element number; }e_{First failed layer }

number/interface of the first failed element; f_{Ultimate failure; }g_{Transverse mode of failure refers to the failure of lamina in a direction perpendicular to the fiber }

irection due to in-plane stresses transverse to fiber direction. d

0 1 2 3 4 5 6 0 20 40 60 80 100 120 140

**(a) Circular cutout**
** First-ply failure**

** Ultimate failure**

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**
**b**
**2/E**
**2**
**h**
**3**

**wmax/h**

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0 15 30 45 60 75

(b) Square cutout

** First-ply failure**
** Ultimate failure**

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**
**b**
**2/E**
**2**
**h**
**3**

**wmax/h**

0.0 0.5 1.0 1.5 2.0 2.5

0 10 20 30 40 50 60

(c) Diamond cutout

** First-ply failure**
** Ultimate failure**

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**
**b**
**2/E**
**2**
**h**
**3**

**wmax/h**

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 15 30 45 60 75

**(d) Elliptical-vertical cutout**
** First-ply failure**

** Ultimate failure**

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**
**b**
**2/E**
**2**
**h**
**3**

**wmax/h**

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0 15 30 45 60 75

**(e) Elliptical-horizontal cutout**
** First-ply failure**

** Ultimate failure**

+ve shear -ve shear
BC1
BC2
BC3
**Nxy**
**b**
**2/E**
**2**
**h**
**3**

**wmax/h**

**Figure 7. Effects of boundary conditions (BC1, BC2 and BC3) on load-deflection response of quasi-isotropic laminate with: (a) **
**Circular; (b) Square; (c) Diamond; (d) Elliptical-vertical; and (e) Elliptical-horizontal cutouts, under positive and negative **
**shear loads. **

**Table 6. Details of failure characteristics of quasi-isotropic laminate with and without various shaped cutouts for boundary **
**conditions BC1, BC2 and BC3 under negative shear load. **

Boundary condition

BC1 BC2 BC3 Cutout

shape

BLa_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/ }

mode of FPF

UFf_{load/ }

mode of UF

BLa_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/mode }

of FPF

UFf

load/mode of UF

BLa_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/ }

mode of FPF UFf

load/mode of UF Without

cutout 51.2/82.0/2.40

21/2/

transverseg _{of stiffness }207.5/loss 77.0/101.4/1.84 _{transverse}21/2/ g 149.8/loss of _{stiffness } 94.2/119.0/1.5_{5} _{transverse}21/2/ g 183.9/loss of _{stiffness }

Circular 28.3/54.8/3.10 37/2/transverse 105.8/loss of _{stiffness } 42.2/62.2/2.20 _{transverse }42/2/ 117.9/loss of _{stiffness } 54.9/65.9/1.38 _{transverse }108/2/ 134.0/loss of _{stiffness }

Square 27.7/44.5/2.32 42/2/transverse _{delamination }66.8/ 41.7/50.7/1.39 108/2/transverse_{delamination}75.2/ 55.5/57.4/0.56 _{transverse }108/2/ _{delamination}78.0/

Diamond 28.5/42.3/2..29

126/interface of layers 15 & 16/delamination

42.3/

delamination 41.9/51.1/1.54

126/interface of layers 15 & 16/delamination

51.1/

delamination -/49.0/-

126/interface of layers 15 & 16/delamination

49/ delamination

Elliptical

vertical 24.0/46.9/2.90 54/2/transverse 47.0/

delamination 37.7/48.1/1.51 54/ 2/transverse

56.3/ delamination43.6/61.1/1.68 54/2/ transverse 62.3/ delamination Elliptical

horizontal 24.7/46.9/2.84

84/interface of layers 14 & 15/delamination

46.9/

delamination 35.3/47.7/1.61

84/interface of layers 14 & 15/delamination

47.7/

delamination45.2/51.1/0.98

84/interface of layers 13 & 14/delamination

51.1/ delamination

a_{Buckling load; }b_{First-ply failure; }c_{Non-dimensionalized maximum transverse deflection at the first-ply failure; }d_{First failed element number; }e_{First failed layer }

number/interface of the first failed element; f_{Ultimate failure; }g_{Transverse mode of failure refers to the failure of lamina in a direction perpendicular to the fiber }

direction due to in-plane stresses transverse to fiber direction.

takes place to cause ultimate failure before buckling. It is
important to note from **Table 5** that under positive shear
load, the effect of cutout shape on first-ply failure load is

condition. For negative shear load, the effect of cutout shape on first-ply failure load is almost uniform for all boundary conditions (maximum variation within 25.6%).

As shown in **Tables 5** and **6**, irrespective of boundary
conditions and loading directions, the modes of first-ply
failure for the laminate with and without square and cir-
cular cutouts remain the transverse mode (*i.e.*, matrix fai-

lure due to stress transverse to fiber direction). However,
the first-ply failure is caused by delamination in the case
of laminate with diamond and elliptical-horizontal
cut-outs with boundary conditions BC2 and BC3, for both
directions of shear load. Variation in first-ply failure mode
with the change in shear load direction (from positive to
negative) can also be observed from matrix failure to de-
lamination and vice-versa, respectively, for the laminate
with diamond and elliptical-horizontal cutouts under
BC1 boundary condition, and for the laminate with an el-
liptical-vertical cutout under BC2 and BC3 boundary con-
ditions. The first-ply failure occurs at the outer edge of
the laminate in the case of the laminate without cutout
for all boundary conditions and for both directions of
shear load. Under positive shear load, the location of the
first-ply failure remains at outer edge of the laminate for
all shaped cutouts for boundary condition BC1, whereas
for other boundary conditions (*i.e.*, BC2 and BC3), the

first-ply failure takes place near the cutout edge for the
laminate with a non-circular cutout. However, in the case
of the laminate with circular cutout, it remains at outer
edge of the laminate. It is important to note from **Table 6**

that under negative shear load, the critical location for the first-ply failure is at cutout edge only for almost all boundary conditions and all shaped cutouts.

From **Tables 5** and **6**, it can be noted that ultimate fail-
ure load also gets reduced with the introduction of a cen-
tral cutout in the laminate. Maximum reduction in ulti-
mate failure load, as compared to the laminate without
cutout, is observed for the laminate with a diamond shap-
ed cutout and minimum reduction is found to be for the
laminate with a circular cutout. This observation is true
for almost all boundary conditions and for both direc-
tions of shear load. It is also important to note that for
positive as well as negative shear load, the effect of cut-
out shape on ultimate failure load is significant for all
boundary conditions. This is based on the fact that the
laminate with a circular cutout bears maximum ultimate
failure load, because the complete loss of stiffness is the
cause of ultimate failure in this laminate; whereas in the
case of laminate with other shaped cutouts, the ultimate
failure load goes to minimum due to early delamination.
It is to be pointed out that the material strength is utilized
efficiently in the case of laminate with a circular cutout
because delamination in the laminate with a circular cut-
out occurs when the laminate has virtually no reserve
strength due to complete loss of stiffness. It is to be noted

that for boundary condition BC1, the effect of cutout shape on ultimate failure load is same (maximum within 60.1%) for positive and negative directions of shear load, whereas for boundary conditions BC2 and BC3, this ef- fect is more (maximum within 72.2% for BC2) for posi- tive shear load than negative shear load (maximum with- in 63.4% for BC3).

It may be noted from **Figures 7(a)-(e)** that except the
values of failure loads (*i.e*., first-ply failure and ultimate

failure loads), the postbuckling response of the laminate
with a cutout in terms of stiffness (given by the slope at a
point in postbuckling range) at a particular value of ma-
ximum deflection is almost the same for boundary condi-
tions BC1, BC2 and BC3 irrespective of cutout shapes
and directions of shear load. So, it can be concluded that
there is no much effect of boundary conditions on post-
buckling stiffness of the laminate with a cutout. It can
also be observed from **Figures 7(a)-(e)** that except in the
case of laminate with an elliptical (vertical and horizontal)
cutout under BC3 boundary condition, although the qua-
si-isotropic laminate with a cutout has more postbuckling
strengths under negative shear load than that under posi-
tive shear load in the early range of postbuckling for va-
rious boundary conditions, but in the advance stage of
postbuckling response, strengths of the laminate become
larger under positive shear load than that under negative
shear load and hence, it can be argued that the laminate
has more postbuckling stiffness under positive shear load
than that under negative shear load. In the case of lami-
nate with an elliptical (vertical and horizontal) cutout un-
der BC3 boundary condition, the load-deflection curves
for positive and negative shear loads (*i.e.*, postbuckling

responses) almost overlap each other and hence, there is no effect of shear load directions on postbuckling stiff- ness. In addition, it is also important to mention that the effects of boundary condition on buckling, first-ply fail- ure and ultimate failure loads (given by maximum per- centage change in buckling, first-ply failure and ultimate failure loads with change in boundary condition) of the quasi-isotropic laminate with a cutout is more for posi- tive shear load than that for the negative shear load for almost all cutout shapes.

**4.2. Effects of Composite Lay-Ups for Various **
**Boundary Conditions **

The effects of composite lay-ups on load-deflection re- sponses of square laminate with a typical central circular cutout under positive and negative shear loads is shown

in **Figures 8(a)-(c)**, respectively, for boundary conditions

0 1 2 3 4 5 6 7 0

20 40 60 80 100

8

**(a) Boundary condition BC1**

+ve shear -ve shear

(45/-45/0/90) 2s (45/-45)

4s (0/90)

4s

**Nxy**

**b**

**2** **/E**

**2**

**h**

**3**

**w**

**max/h**

(a)

0 1 2 3 4 5 6 7

0 20 40 60 80 100 120 140

8

**(b) Boundary condition BC2**

+ve shear -ve shear

(45/-45/0/90) 2s (45/-45)

4s (0/90)

4s

**Nxy**

**b**

**2** **/E**

**2**

**h**

**3**

**w**

**max/h**

(b)

0 1 2 3 4 5 6 7 8

0 20 40 60 80 100 120 140

**(c) Boundary condition BC3**

+ve shear -ve shear

(45/-45/0/90)_{2s}
(45/-45)_{4s}
(0/90)_{4s}
**Nxy**

**b**

**2** **/E**

**2**

**h**

**3**

**w _{max}/h**

(c)

**Figure 8. Effect of composite lay-ups on load-deflection re- **
**sponse of square laminate with a typical circular cutout un- **
**der positive and negative shear loads, subjected to bound- **
**ary conditions: (a) BC1; (b) BC2; and (c) BC3. **

are considered for this study. Corresponding details of
failure characteristics for these laminates under positive
and negative directions of shear loads and boundary con-
ditions BC1, BC2 and BC3 are given in** Table 7**. It can
be noted from **Table 7** that for BC1, angle-ply [*i.e.*,

(45/−45)4s] and cross-ply [*i.e.*, (0/90)4s] laminates have
maximum and minimum buckling strengths, respectively,
for both directions of shear load. However, for other two
boundary conditions (*i.e.*, BC2 and BC3), cross-ply and

angle-ply laminates have maximum and minimum buck-
ling strengths, respectively, for positive as well as nega-
tive shear load. It is also to be observed that for BC3
boundary condition, the onset of buckling is precluded
for (0/90)4s laminate under negative shear load since the
first-ply failure and subsequently the ultimate failure
(due to complete loss of stiffness) because of the succes-
sive failures at the same load occur before buckling. Ir-
respective of shear directions and boundary conditions,
(+45/−45/0/90)2s laminate has the maximum first-ply fai-
lure load, whereas the (45/−45)4s laminate has minimum
first-ply failure load. From **Table 7**, it is also to notice
that almost all laminate configurations have more buck-
ling and first-ply failure loads under negative shear load
than that under positive shear load, irrespective of boun-
dary conditions.

As presented in **Table 7** that except for (0/90)4s lami-
nate with boundary condition BC1 under negative shear
load, the mode of first-ply failure for all laminate confi-
gurations remains matrix failure (*i.e.*, transverse mode of

**Table 7. Detail of failure characteristics of (+45/−45/0/90)2s, (45/−45)4s and (0/90)4s laminates with a typical circular cutout **

**under positive and negative shear loads for boundary conditions BC1, BC2 and BC3. **

Positive shear Negative shear
Laminate _{conditions }Boundary _{BL}a_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/mode of }

FPF

UFf_{ load/mode }

of UF

BLa_{/FPF}b

load/(*w*max/*h*)c

FEd_{/FL}e_{/mode of }

FPF

UFf_{ load/mode }

of UF

BC1 20.7/39.3/2.07 139/1/transverse 101.9/loss of _{stiffness } 28.0/54.8/3.10 37/2/transverse 105.8/loss of _{stiffness }

BC2 32.7/56.5/2.03 1/16/transverse 132.1/loss of _{stiffness } 42.2/62.2/2.20 42/2/transverse 117.9/loss of _{stiffness }

(45/

−

45/0/9

0)2s

BC3 43.9/69.2/1.85 133/1/transverse 119.0/loss of _{stiffness } 54.9/65.9/1.38 108/2/transverse 134.0/loss of _{stiffness }

BC1 23.0/31.5/2.08 72/16/transverse 59.4/loss of _{stiffness } 28.2/38.0/2.48 42/2/transverse 65.0/loss of _{stiffness }

BC2 31.0/37.3/1.49 72/1/transverse 69.2/loss of _{stiffness } 36.8/41.8/1.48 42/2/transverse 69.4/loss of _{stiffness }

(45/

−

45)

4s

BC3 37.4/42.1/1.07 72/16/transverse 83.6/loss of _{stiffness } 43.7/46.6/0.94 42/15/transverse 78.9/loss of _{stiffness }

BC1 15.7/35.0/2.00 1/1/transverse 62.4/loss of _{stiffness } 19.0/46.0/2.82 103/1/ in-plane _{shear } 56.5/loss of _{stiffness }

BC2 36.0/51.1/1.52 1/16/transverse 110.6/loss of _{stiffness } 43.5/58.5/1.71 1/1/ transverse 92.5/loss of _{stiffness }

(0

/90)

4s

BC3 58.2/58.4/0.20 139/2/transverse 120.7/loss of _{stiffness } -/58.5/- 1/1/ transverse 58.5/loss of _{stiffness }

a_{Buckling load; }b_{First-ply failure; }c_{Non-dimensionalized maximum transverse deflection at the first-ply failure; }d_{First failed element number; }e_{First failed layer }

number/interface of the first failed element; f_{Ultimate failure; }g_{Transverse mode of failure refers to the failure of lamina in a direction perpendicular to the fiber }

direction due to in-plane stresses transverse to fiber direction.

It is also worth mentioning that as in the case of quasi-
isotropic laminate with various shaped cutout, the effects
of boundary condition on buckling, first-ply failure and
ultimate failure loads for the angle-ply and cross-ply la-
minates with a circular cutout is maximum for positive
shear load than the negative shear load. Furthermore, it is
noted that for both directions of shear load, the maximum
effects of boundary condition on buckling and failure
loads is for cross-ply laminate, whereas the minimum ef-
fects is observed for angle-ply laminate. Moreover, it is
also important to note from **Figures 8(a)-(c)**, the post-
buckling stiffness (given by the slope of load versus de-
flection curve at a particular value of maximum trans-
verse deflection) of (0/90)4s laminate is maximum, where-
as it is minimum for (45/−45)4s laminate, irrespective of
boundary conditions and directions of shear load.

**5. Concluding Remarks **

Based on the aforementioned results and discussion, the following important observations can be made:

The effect of cutout shapes on buckling load of qua- si-isotropic laminate is more for BC2 and BC3 boun- dary conditions, respectively, under positive and ne- gative directions of shear load.

The effect of boundary conditions on postbuckling stiffness is not appreciable for quasi-isotropic lami-

nate with a cutout of various shapes.

For boundary condition BC1, angle-ply [*i.e.*, (45/

−45)4s] and cross-ply [*i.e.*, (0/90)4s] laminates have
maximum and minimum buckling strengths, respec-
tively, for both directions of shear load. However, for
other two boundary conditions, reverse is true for po-
sitive as well as negative shear load.

All laminate configurations have more buckling and first-ply failure loads under negative shear load than that under positive shear load, irrespective of bound- ary condition.

Matrix failure due to transverse stresses is the pre- dominant mode of first-ply failure for various lami- nate configurations under positive as well as negative shear load for all boundary conditions.

Irrespective of shear directions and boundary condi- tions, the quasi-isotropic laminate has the maximum first-ply failure and ultimate failure loads as compa- red to the angle-ply and cross-ply laminates.

Irrespective of laminate configurations, the laminate with a circular cutout has more effects of boundary conditions on buckling, first-ply failure and ultimate failure loads under positive shear load than that under the negative shear load; the maximum and minimum effects being for the cross-ply and the angle-ply la- minates, respectively.

more postbuckling stiffness under positive shear load than the laminate under negative shear load, irrespec- tive of boundary condition; and the cross-ply and an- gle-ply laminates have maximum and minimum post- buckling stiffness, respectively.

**REFERENCES **

[1] R. M. Jones, “Mechanics of Composite Materials,” McGraw- Hill, Kogakusha Ltd., Tokyo, 1975.

[2] S. B. Singh and D. Kumar, “Postbuckling Response and
Failure of Symmetric Laminated Plates with Rectangular
Cutouts under Uniaxial Compression,” *Structural Engi- *
*neering and Mechanics*, Vol. 29, No. 4, 2008, pp. 455-

467.

[3] T. Özden, “Analysis of Critical Buckling Load of Lami-
nated Composites Plate with Different Boundary Condi-
tions Using FEM and Analytical Methods,” *Computa- *
*tional Materials Science*, Vol. 45, No. 4, 2009, pp. 1006-

1015. doi:10.1016/j.commatsci.2009.01.003

[4] S. K. Panda and L. S. Ramachandra, “Buckling of Rec-
tangular Plates with Various Boundary Conditions Load-
ed by Non-Uniform In-Plane Loads,” *International Jour- *
*nal of Mechanical Sciences*, Vol. 52, No. 6, 2010, pp.

819-828. doi:10.1016/j.ijmecsci.2010.01.009

[5] D. Kumar and S. B. Singh, “Effects of Boundary Condi-
tions on Buckling and Postbuckling Responses of Com-
posite Laminate with Various Shaped Cutouts,” *Compos- *
*ite Structures*, Vol. 92, No. 3, 2010, pp. 769-779.

doi:10.1016/j.compstruct.2009.08.049

[6] S. B. Singh and D. Kumar, “Postbuckling Response and
Failure of Symmetric Laminated Plates with Rectangular
Cutouts under In-Plane Shear,” *Structural Engineering *
*and Mechanics*, Vol. 34, No. 2, 2010, pp. 175-188.

[7] Y. Zhang and F. L. Matthews, “Postbuckling Behaviour
of Anisotropic Laminated Plates under Pure Shear and
Shear Combined with Compressive Loading,”* Journal of *
*American Institute of Aeronautics and Astronautics*, Vol.

22, No. 2, 1984, pp. 281-286.

[8] S. B. Singh and A. Kumar, “Postbuckling Response and Failure of Symmetric Laminates under In-Plane Shear,”

*Composite Science and Technology*, Vol. 58, No. 12,

1998, pp. 1949-1960.

doi:10.1016/S0266-3538(98)00032-3

[9] M. P. Nemeth, “Buckling and Postbuckling Behavior of Laminated Composite Plates with a Cutout,” 1996. [10] V. O. Britt, “Shear and Compression Buckling Analysis

for Anisotropic Panels with Elliptical Cutouts,”* Journal *
*of American Institute of Aeronautics and Astronautics*,

Vol.32, No. 11, 1994, pp. 2293-2299.

[11] R. J. Herman, “Postbuckling Behavior of Graphite/Epoxy
Cloth Shear Panels with 45˚-Flanged Lightening Holes,”
M.S. Thesis, Naval Postgraduate School, Monterey, 1982.
[12] G. J. Turvey and K. Sadeghipour, “Shear Buckling of Ani-
sotropic Fibre-Reinforced Rectangular Plates with Central
Circular Cutouts,” *Proceedings of the International Con- *
*ference Computer Aided Design in Composite Material *
*Technology*, Southampton, 1988, pp. 459-473.

[13] S. Vellaichamy, B. G. Prakash and S. Brun, “Optimum De- sign of Cutouts in Laminated Composite Structures,”

*Computers & Structures*, Vol. 37, No. 3, 1990, pp. 241-
246. doi:10.1016/0045-7949(90)90315-S

[14] P. N. Jha and A. Kumar, “Response and Failure of Square
Laminates under Combined Loads,” *Composite Structures*,

Vol. 55, No. 3, 2002, pp. 337-345. doi:10.1016/S0263-8223(01)00141-6

[15] P. Jain and A. Kumar, “Postbuckling Response of Square
Laminates with a Central Circular/Elliptical Cutout,” *Com- *
*posite Structures*, Vol. 65, No. 2, 2004, pp. 179-185.
doi:10.1016/j.compstruct.2003.10.014

[16] S. J. Guo, “Stress Concentration and Buckling Behaviour
of Shear Loaded Composite Panels with Reinforced Cut-
outs,”* Composite Structures*, Vol. 80, No. 1, 2007, pp. 1-9.

doi:10.1016/j.compstruct.2006.02.034

[17] S. Guo,R. Morishima, X. Zhang and A. Mills, “Cutout
Shape and Reinforcement Design for Composite C-Sec-
tion Beams under Shear Load,” *Composite Structures*,
Vol. 88, No. 2, 2009, pp. 179-187.

doi:10.1016/j.compstruct.2008.03.001

[18] D. Kumar and S. B. Singh, “Postbuckling Strengths of Com-
posite Laminate with Various Shaped Cutouts under In-
Plane Shear,”* Composite Structures*, Vol. 92, No. 12,
2010, pp. 2966-2978.

doi:10.1016/j.compstruct.2010.05.008

[19] O. O. Ochoa and J. N. Reddy, “Finite Elements Analysis of Composite Laminates,” Kluwer Academic Publishers, Dordrecht, 1992.

[20] S. B. Singh, “Postbuckling Response, Strength and Fail- ure of Symmetric Laminates,” Ph.D. Dissertation, Indian Institute of Technology, Kanpur, 1996.

[21] S. Kosteletos, “Postbuckling Response of Laminated Plates under Shear Loads,”Composite Structures, Vol. 20, No. 3, 1992, pp. 137-145. doi:10.1016/0263-8223(92)90020-D [22] Z. Hashin, “Failure Criteria for Unidirectional Fiber Com-

posites,”* Journal of Applied Mechanics*, Vol. 47, No. 3,

1980, pp. 329-334. doi:10.1115/1.3153664

[23] S. W. Tsai, “A Survey of Macroscopic Failure Criteria for
Composite Materials,”* Journal of Reinforced Plastics and *
*Composites*, Vol. 3, No. 1, 1984, pp. 40-62.

**Appendix **

**(1) Tsai-Hill Criterion **

In this criterion a considerable interaction exists among failure strengths of the lamina as against the other non- interactive criteria such as Hashin [22] and Tsai’s [23] tensor polynomial criteria. Tensor polynomial form of the Tsai-Hill criterion can be obtained from the following most general polynomial failure criterion of Tsai [23] at failure state.

1 1 2 2 3 3 12 1 2 13 1 3 23 2 3

2 2 2 2 2 2

11 1 22 2 33 3 44 23 55 13 66 12

2 2 2

1

*F* *F* *F* *F* *F* *F*

*F* *F* *F* *F* *F* *F*

wherein, *Fi*,* Fij*are the strength tensors of the second and

fourth rank; 1, 2, 3_{ are the normal stress components }

in principal material directions 1, 2 and 3, respectively (the subscript 1 referring to the fiber direction);

12, 13, 23

are the shear stress components in the planes 1 - 2, 1 - 3 and 2 - 3, respectively. In Tsai-Hill criterion the following strength tensors are used in the above ex- pression.

1 2 3 0;

*F* *F* *F* *F*_{11} 1_{2};*F*_{22} 1_{2};*F*_{33} 1_{2};

*X* *Y* *Z*

44 2 55 2 66 2

1 1 1

; ; ;

*F* *F* *F*

*R* *S* *T*

12 2 2 2

1 1 1 1 _{;}

2

*F*

*X* *Y* *Z*

_{} _{}

13 2 2 2

1 1 1 1 _{;}

2

*F*

*Z* *X* *Y*

_{} _{}

23 2 2 2

1 1 1 1

; 2

*F*

*Y* *Z* *X*

_{} _{}

In the above expressions *X*,* Y *and* Z *are the normal

strengths (tensile or compressive, depending upon the
sign of _{1}, _{2}, _{3}) along principal material directions 1,
2 and 3, respectively; *R*,* S and T* are the shear strengths

of lamina in planes 2 - 3, 1 - 3 and 1 - 2, respectively

**(2) Interlaminar Failure Friterion **

As per the interlaminar failure criterion, the onset of de- lamination takes place when the interlaminar transverse stress (calculated by integration of equilibrium equations) components satisfy the following expression:

2 _{2} _{2}

3 13 23

2 1

*DN* *DS*

_{} _{}

where, 3 is the transverse normal stress component; 13, 23

are the transverse shear stress components in principal material planes 1 - 3 and 2 - 3, respectively;

*DN*

is the peel strength and *DS* is the interlaminar